= Algebraic torsion of a Dirac field
{title2=$K_{abc}=-\kappa^2\bar\chi\gamma_{abc}\chi/4$}
Use $K_{abc}=e_a{}^\mu(\omega-\omega(e))_{\mu bc}$ and $T_{abc}=K_{bac}-K_{cab}$. For the symmetrized Dirac <action> $-e\bar\chi\gamma^\mu\overleftrightarrow\nabla_\mu\chi/2-em\bar\chi\chi$, varying the <spin connection> gives $-e\bar\chi\gamma^{\mu ab}\chi\,\delta\omega_{\mu ab}/4$. Only totally antisymmetric contorsion is sourced. Its <action> is $-eK^2/(2\kappa^2)-eK_{abc}\bar\chi\gamma^{abc}\chi/4$, hence the stated solution. Substitution generates $e\kappa^2(\bar\chi\gamma^{abc}\chi)(\bar\chi\gamma_{abc}\chi)/32$. The torsion is an <auxiliary field>, not an independent propagating degree of freedom.
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